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In set theory, a tree is a partially ordered set (T, <) such that for each t ∈ T, the set {s ∈ T : s < t} is well-ordered by the relation <. Frequently trees are assumed to have only one root (i.e. minimal element), as the typical questions investigated in this field are easily reduced to questions about single-rooted trees.
Every tree in descriptive set theory is also an order-theoretic tree, using a partial ordering in which two sequences and are ordered by if and only if is a proper prefix of . The empty sequence is the unique minimal element, and each element has a finite and well-ordered set of predecessors (the set of all of its prefixes). An order-theoretic ...
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, [1] except for the root node, which has no parent (i.e., the root node ...
A tree structure, tree diagram, or tree model is a way of representing the hierarchical nature of a structure in a graphical form. It is named a "tree structure" because the classic representation resembles a tree, although the chart is generally upside down compared to a biological tree, with the "stem" at the top and the "leaves" at the ...
Kurepa tree. In set theory, a Kurepa tree is a tree (T, <) of height ω 1, each of whose levels is countable, and has at least ℵ 2 many branches. This concept was introduced by Kurepa (1935). The existence of a Kurepa tree (known as the Kurepa hypothesis, though Kurepa originally conjectured that this was false) is consistent with the axioms ...
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. . More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ (so Aronszajn trees are the same as -Aronszajn tree
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
Monadic second-order logic. In mathematical logic, monadic second-order logic ( MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification over sets. [ 1] It is particularly important in the logic of graphs, because of Courcelle's theorem, which provides algorithms for evaluating monadic second ...